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22.3 Well-Formed Formulas vs. Sentences

You will remember from our first logical system, BOOL, that sentences play a key role in logic: they are the constituents of arguments, and each one is true or false.

FOL is bivalent, just like BOOL and PROP. We want every sentence of FOL to have exactly one truth value: T or F.

But what about Dog(x)? When x is a free variable, we said that this formula doesn’t have a truth value, because it doesn’t yet make a determinate claim about the world.

What that means is that Dog(x) is not a sentence.

What we need is a new grammatical category for Dog(x): we will call it an open formula, because it has a free variable in it.

Open formula: has free variable
Closed formula: sentence

If a formula has no free variable, then it is a closed formula, in other words, a sentence.

Open and closed formulas are different, but there is still something they share: they both have a kind of complete “sentence-like” structure.

For example, Likes(x, ) is not an open formula, because it’s not even grammatically complete yet. Likes() is a binary predicate, and Likes(x, ) doesn’t even have the right number of terms.

Here’s what open and closed formulas share: they both have predicates with the right number of terms, and are in all ways grammatically correct. The only difference is that open formulas have free variables, whereas closed ones don’t.

Well-Formed Formula (wff): open and closed formulas

The general term we’ll use to encompass both is well-formed formulas, or wffs. This might sound silly, but logicians pronounce “wff” as WOOF.

Have your laughs now; soon you’ll be used to it too!